Numerical solution of parabolic equations in high dimensions
Petersdorff, Tobias Von ; Schwab, Christoph
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 38 (2004), p. 93-127 / Harvested from Numdam

We consider the numerical solution of diffusion problems in (0,T)×Ω for Ω d and for T>0 in dimension d1. We use a wavelet based sparse grid space discretization with mesh-width h and order p1, and hp discontinuous Galerkin time-discretization of order r=O(logh) on a geometric sequence of O(logh) many time steps. The linear systems in each time step are solved iteratively by O(logh) GMRES iterations with a wavelet preconditioner. We prove that this algorithm gives an L 2 (Ω)-error of O(N -p ) for u(x,T) where N is the total number of operations, provided that the initial data satisfies u 0 H ϵ (Ω) with ϵ>0 and that u(x,t) is smooth in x for t>0. Numerical experiments in dimension d up to 25 confirm the theory.

@article{M2AN_2004__38_1_93_0,
     author = {Petersdorff, Tobias Von and Schwab, Christoph},
     title = {Numerical solution of parabolic equations in high dimensions},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     volume = {38},
     year = {2004},
     pages = {93-127},
     doi = {10.1051/m2an:2004005},
     mrnumber = {2073932},
     zbl = {1083.65095},
     language = {en},
     url = {http://dml.mathdoc.fr/item/M2AN_2004__38_1_93_0}
}
Petersdorff, Tobias Von; Schwab, Christoph. Numerical solution of parabolic equations in high dimensions. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 38 (2004) pp. 93-127. doi : 10.1051/m2an:2004005. http://gdmltest.u-ga.fr/item/M2AN_2004__38_1_93_0/

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